engineering mathematics education with computer algebra

ENGINEERING MATHEMATICS EDUCATION WITH COMPUTER ALGEBRA: THE
MATLAB ALTERNATIVE
James Katende
Faculty of Technology
Bayero University, Kano
Keywords: Computer Algebra Systems, Engineering Mathematics Education
Abstract
Computer algebra systems have become an
important tool for many engineering and
technical professionals. There is a growing need
to incorporate such tools into the education of
such professionals. This paper discusses these
systems and their role within engineering
mathematics in higher education. Some
advantages and problems associated with
computer algebra are highlighted and illustrated
using MATLAB.
1. Introduction
In the early 1980s there were predictions that
computer algebra systems (also called computer
based symbolic manipulation systems and
abbreviated to SAMs) would bring about a
revolution. Stern [1] asserted that SAMs would
do for mathematical analysis what the hand-held
calculator had done for calculation and
arithmetic. However, the revolutionary change to
practice in higher education that many have
predicted has not yet happened. Lawson [2]
suggested two main reasons for this: one related
to teaching staff and the other to students. The
first is the innate conservatism of many
mathematicians in higher education coupled with
the fact that there were a number of failings in
most of the early packages [3].
The second reason is one of access. Students
readily use pocket calculators because of
familiarity and access. The same has not been
true of SAMs. Students have to visit computer
laboratories in order to be able to use a SAM.
Consequently, SAMs have not yet shared the
universal availability, which the calculator
enjoys.
However, SAMs are an increasingly important
tool for professionals in a wide variety of
technical and scientific occupations. It was
recognized early that those responsible for
designing and providing the education, both
initial formation and continuing professional
development, of these professionals must ensure
that they acquire appropriate knowledge of the
capabilities and limitations of such systems and
learn to incorporate their use into the overall
pattern of their professional activities [4]
1
This paper briefly reviews what a SAM is, and
outlines some of the potentials and problems
which attend the use of SAMs in undergraduate
engineering mathematics. Illustrations are
presented using the programming language
MATLAB whose Symbolic Math Toolbox
utilizes tools built on portions of one of the most
popular SAMs, MAPLE.
3. SAMs and Their Uses
Basically, a SAM is a piece of software, which is
capable of working symbolically as well as
numerically. In principle it is a program that does
on a computer the manipulation that has
traditionally been done with pencil and paper.
So, for example, if requested to expand cos(7x) a
SAM will effortlessly return the answer:
64cos 7 (x)-112cos 5 (x)+56cos 3 (x)-7cos(x)
Similarly, a SAM command to differentiate the
function sin(E x /(x 2 +1) with respect to x produces
the answer (almost instantly):
x ⎛ x
⎛ E ⎞
⎜ E
cos ⎜
⎟ − 2
2
⎝ + ⎠
⎜ 2
x 1
⎝
x + 1
E
x
x
⎞
( ) ⎟⎟
2 2
x + 1
Early SAMs such as, MACSYMA and
REDUCE, were large, required considerable
computer power and had complex syntax.
However, hardware and software advances over
recent years completely altered this situation.
Modern programs that incorporate SAM
capabilities include Derive, Maple, Mathematica
and Matlab. They run on PCs, have graphical
user interfaces and although there is still a
reasonable amount of syntax required for more
complicated operations, can be used to perform
routine manipulation without the need for
lengthy study of manuals. In addition these
packages have progressed enormously beyond
simply being manipulators of complicated
symbolic strings. They are now complete
mathematical environments with algebraic,
numerical, graphical and programming facilities.
⎠

3. The MATLAB SAM
MATLAB (Matrix Laboratory) is a high
performance interactive software tool for
technical computing. Typical uses include
general-purpose numeric computation; algorithm
prototyping; modelling and simulation; data
analysis, exploration and visualisation; scientific
and engineering graphics; a teaching aid in linear
algebra and other topics; and solution of special
purpose matrix formulations such as those
associated with automatic control, statistics and
signal processing [5,6,7].
MATLAB features a family of applicationspecific
solutions called toolboxes, which allow
the user to learn and apply specialised
technology. Toolboxes are specialised
collections of MATLAB functions (M-files) that
extend the MATLAB environment to solve
particular classes of problems. Areas in which
toolboxes are available include: symbolic maths;
analogue and digital signal processing; analogue
and digital control systems; neural networks;
fuzzy logic; simulation; system identification;
and many others.
The Symbolic Toolbox deals with symbols,
formulas, and equations. It features in-built
functions for: differentiation; integration;
solution of differential equations; Laplace and
Fourier transformations; linear algebra; graphics;
etc. It is employed in the following sections to
illustrate some advantages and drawbacks
associated with use of a SAM in mathematics
education.
Clicking the MATLAB (version 5.3) icon in the
WINDOWS – 95/98/NT/Me environment opens
the command window, which is the main
window for communicating with the MATLAB
interpreter. The command window prompt is "»".
To find out about the symbolic computation
functions available in MATLAB, execute the
command:
» help symbolic.
4. Uses of SAMs in Education
A strong case has been presented for the
widespread use of computer algebra in
mathematics education. Some benefits resulting
from using computer algebra have been
enumerated [8]. The key advantages include the
following:
Computer algebra allows the focus to be
on concepts and understanding
principles without the distraction of
2
large amounts of ‘routine’
manipulation.
Motivation can be enhanced because
more realistic problems can be tackled.
Students can cover many more
examples allowing them to be more
active participants in discovery
learning.
These are expanded in the following with
illustrative examples in MATLAB. Although all
the examples of computer algebra commands
given in this section are for MATLAB, all SAM
packages can easily do the things shown here.
Where MATLAB sessions are shown, lines
beginning with '»' show inputs to MATLAB and
other lines show answers MATLAB returns. The
bold typeface shows MATLAB in-built
functions or constants.
(a) SAMS reduce the premium on pure
manipulative ability. Students may use a SAM to
assist in or confirm their own calculations. As a
result weaker students can be enabled to follow
more sophisticated or complex arguments. In the
following MATLAB session a student enters the
expression, f = sin (yx 2 ), and differentiates it
with respect to x and y in turn.
»f = sym ( 'sin(y*x^2)');
»diff (f) % derivative with respect to x
ans =
2*cos(y*x^2)*y*x
»diff(f,'y') %derivative with respect to y
ans =
cos(y*x^2)*x^2
(b) SAMs enable students to concentrate on
concepts, not detail. A case in point is that of
differentiation. In undergraduate engineering
mathematics, students learn a range of
techniques of differentiation including such
things as the product rule, the quotient rule, and
the function of a function rule, implicit
differentiation and logarithmic differentiation. A
SAM will return the derivative of a function
without any explicit indication that a particular
technique or rule has been used. For example,
differentiation of the function in the above
MATLAB session requires use of both the
function of a function and the product rule.
However, as far as the user of a SAM is
concerned, there is no difference between
differentiating this function and differentiating x 2
as illustrated in the following MATLAB session.
Note that in both cases MATLAB returns the
answer (ans) almost instantaneously.

»x=sym ('x');
»diff(x^2)
ans =
2*x
This can be viewed as a great advantage as it
makes the point that differentiation is the same
process no matter how complex the function
being differentiated. The technique employed to
determine the derivative does not alter what the
derivative means.
Consequently students need only to know that
there is a product rule, what the rule is and where
it comes from but need not necessarily complete
large numbers of ever more (algebraically)
complicated examples. In the time that is freed
by this reduction in “drill” exercises, key
concepts such as the derivative being a rate of
change, qualitative information such as a positive
derivative indicating an increasing function and
using derivatives to represent physical quantities
(such as velocity and acceleration) are all topics
which might be more heavily emphasized than at
present.
(c) A SAM acts as a computational assistant for
routine procedures. Once a procedure has been
understood at a basic level a better understanding
can sometimes be achieved by the study of a
range of examples. However, the labour of
repetitive computations often inhibits such
learning. A SAM can reduce the labour and also
allow lecturers to set a wider range of illustrative
work in assignments. Fourier series offer a good
example. To illustrate the more subtle points of
the topic requires a wide range of examples to be
completed. The labour of calculating the Fourier
coefficients for that large number of examples
discourages students from completing them and
thus inhibits understanding. The following
MATLAB session evaluates the coefficients bn
for the first five terms of a Fourier series for the
function f(x) = x(1-x), 0≤x≤1. The Fourier
coefficients are given by:
1
bn = x ( 1 x)
sin( πnx)
dx
∫ −
0
With pencil and paper this would require
integration by parts twice.
» syms x n bn
»bn = int(x*(1-x)*sin(pi*n*x),0,1)
bn =
-(pi*n*sin(pi*n)+2*cos(pi*n))/pi^3/n^3+
3
+2/pi^3/n^3
» pretty(bn)
pi n sin(pi n) + 2 cos(pi n) 2
- ---------------------------------- + ------
3 3 3 3
pi n pi n
» subs(bn,n,{1,2,3,4,5})
bn =
[ 4/pi^3, 0, 4/27/pi^3, 0, 4/125/pi^3]
The result is actually
πnsin(
πn)
+ 2cos(
πn)
2
bn = −
+
3 3
3
π n π n
n
( −1)
2
= − 2 + 3 3 3 3
π n π n
The first five coefficients are:
bn = [
4
3
π
, 0,
4
27π
3
, 0,
4
125
3
π
Majority of engineering undergraduates would
make at least one minor error in completing the
above integration. This would simply obscure
any learning, which was intended to follow from
a consideration of the (correct) Fourier series
representation of the waveform. Using a SAM,
students can be expected to complete a wider
range of examples more quickly and more
accurately, and when guided appropriately, to
gain thus an enhanced understanding of the
topic.
(d) SAMs act as an expert system. A student may
know, in principle, that a manipulation is
possible but may not have the knowledge, skill
or persistence to carry it through in practice.
Trigonometric identities provide good examples
of this. Having met the identities for cos(2x) and
cos(3x) as polynomials in cos(x), a student might
perhaps know, in theory, that cos(6x) could also
be expressed as a polynomial in cos(x) but be
unable to complete the computation quickly or
accurately. Below is a MATLAB session
showing how this may be done with a SAM.
» x=sym ('x');
» expand (cos(2*x))
ans =
2*cos(x)^2-1
]
3

» expand (cos(3*x))
ans =
4*cos(x)^3-3*cos(x)
» expand (cos(6*x))
ans =
32*cos(x)^6-48*cos(x)^4+18*cos(x)^2-1
Note that MATLAB’s response is almost
instantaneous.
(e) SAMs make graph drawing and visualisation
easier. The sophisticated graph plotting facilities
of most SAMs can help to give insight into
mathematical ideas through visualisation. For
example the function f(x) =x(1-x) in (c) may be
visualised using the command:
» ezplot('x*(1-x)',0,1)
The resulting plot is shown in Fig.1.
0.25
0.2
0.15
0.1
0.05
0
x*(1-x)
0 0.1 0.2 0.3 0.4 0.5
x
0.6 0.7 0.8 0.9 1
Fig. 1: A visualisation of the function whose
Fourier coefficients were obtained in (c).
Three- dimensional visualizations are supported.
The following example illustrates the creation of
a 3-D plot of the sinc function, sin(r)/r.
» [X,Y]=meshgrid(-9:.5:9);
» R=sqrt(X.^2+Y.^2)+eps;
» Z=sin(R)./R;
» mesh(X,Y,Z)
The plot is shown in Fig. 2. In this example, R is
the distance from the origin, which is the centre
of the matrix. Adding eps avoids the
indeterminate 0/0 at the origin.
4
Fig 2. 3-D visualisation of the sinc function
(f) SAMs encourage exploration and experiment.
The many features of SAMs can be used to
encourage students to be more open to
mathematical exploration and experimentation.
The SAM removes the drudgery of repetitive
calculation, provides standard tools, eliminates
much of the need for routine reference to tables
and textbooks and generally makes
experimentation easier, more exciting and more
rewarding.
(g) Lecturers can construct complex/animated
visualisations and improve instruction. The
sophisticated graphical features of many SAMs
engender the construction of visualisations and
animations, which can motivate or clarify
complex mathematical ideas. For example a
SAM may be used as a virtual laboratory where
students perform computer-based experiments on
animated mass-spring-damper systems [8]. They
can then relate the results of their virtual
experiments to solutions of the differential
equations that model such systems.
5. Some problems with use of a SAM
(a) Additional time is needed to learn SAM
usage. This would inevitably take time from
other subjects in the curriculum. However, skill
with a SAM will ultimately improve the student's
efficiency in learning other material, so that part,
at least, of the time spent on learning a SAM
would be recovered.
(b) Undermining development of manipulative
skills. Just as the pocket calculator is considered
to have led to the atrophy of arithmetic skills, so
might SAMs probably lead to a reduction in
purely manipulative algebra and calculus skills?
However, once SAMS become available on
affordable pocket calculators, it may be argued
that this loss is unimportant. Indeed Lawson [9]
argues that computer algebra ought not to be
seen as a de-skilling tool but rather as a re-

skilling one that gives access to higher level
skills (such as formulation and interpretation)
instead of being focussed on lower level skills of
manipulation and techniques.
(c) Problems of remembering the SAM language.
The rich vocabulary of SAM commands contains
a number of possibilities. To become truly adept
in a SAM language requires routine everyday
use. Occasional users would quickly lose their
'edge'. The following MATLAB session
illustrates what can easily happen in a SAM if an
occasional user attempts to combine sin(A+B)
and sin(A-B). The student has an idea of the
form the answer takes but wants MATLAB to
confirm the correct details. Among the many
related commands of MATLAB which one is
correct? The approach adopted here is to try any
possibility until the correct answer is obtained.
Obviously this can be time consuming and
frustrating for a novice or occasional user. Note
that the command simple (not used here) gives
even intermediate steps used in the process.
» syms a b x
» x=sin(a+b)+sin(a-b)
x =
sin(a+b)+sin(a-b)
» simplify (x)
ans =
sin(a+b)+sin(a-b)
» factor (x)
ans =
sin(a+b)+sin(a-b)
» expand (x)
ans =
2*sin(a)*cos(b)
(d) Course examination. Examination of a SAMbased
course is a serious problem. With the everincreasing
mathematics classes, it is impractical
to consider the option of providing a
microcomputer at each exam desk. All forms of
non-examination assessment (e.g. continuous
assessment only) might not be acceptable,
especially in a society in which traditional values
of 'fair play' and honesty can no longer be taken
for granted.
6. Conclusion
In the face of the demonstrated versatility of
SAMs there is a need to change either the
engineering mathematics curriculum or the
approach to the curriculum in such a way as to
exploit the numerous advantages they (SAMs)
offer. Engineering mathematics teachers should
5
not only use SAMs as a tool to free students
from the drudgery of routine algebra, calculus
and graph drawing, but should deploy SAMs to
help them (students) to understand at a deeper
level, the nature and structures of mathematics
and to appreciate the ways SAMs can be used to
model the operation of those physical and other
real-world systems that are of interest to
engineers and technologists.
7. References
1. Stern, L.A.: "Computer calculus",
Science News, 1981, 119, Reprinted in
SIGSAM Bull, 1981, 15, (3), pp.26-27
2. Lawson, D.: "The challenge of
computer algebra to engineering
mathematics", Engineering Science.
and Education Journal, 1997, 6, (6),
pp.228-232.
3. Mitic, P. and Thomas, P.G.: "Pitfalls
and limitations of computer algebra",
Computers and Education, 1994, 22,
(4), pp.355-361.
4. Barozzi, G., and Clements, R. R.: "The
potential use of computer algebra
systems in mathematical education of
engineers", Int. J. Math. Educ. Sci.
Tech., 1987, 18, p.681.
5. Chipperfield, A.J. and Fleming, P.J.
(Eds.) 'MATLAB toolboxes and
applications for control', Peter
Peregrinus/IEE, UK, 1993.
6. Biran, A. and Breiner, M. 'MATLAB
for Engineers', Addison-Wesley,
Harlow, England, 1995.
7. http://www.mathworks.com: MATLAB
the Language for Technical Computing:
Visualisation, Programming, 1998.
8. Clements, R.R: "Computer algebra for
engineers: the Maple alternative",
Engineering Science. and Education
Journal, 1997, 6, (6), pp.233-238.
9. Rozsasi, R.: "The use of Mathematica
programmed virtual experiment
animations", in Hibberd, S. and Mustoe,
L. (Eds.): 'Mathematical Education of
Engineers' (IMA Publications, 1997)